Multiplicative orders modulo powers

[Dimiter Skordev]

Let a and b be relatively prime integers greater than 1. Is it possible for the multiplicative orders of a modulo two different powers of b to be equal? I am especially interested in the case of a = 2.

 

[Dimiter Skordev]

The instance with a = 2 was posted also in MathOverflow. A positive answer was given there with any Wieferich prime number as b, the relevant powers being this number itself and its square. Is it possible an example where both powers are other than b?